One of the fundamental problems in contact topology is to classify contact structures on a given 3-manifold. In particular, classifying contact structures on surgeries along a given knot has been very poorly studied. The only fully understood case so far is that of the unknot (lens spaces); for all other knots we have only partial results, or none at all. Several topological and algebraic tools have been developed to attack this problem.
In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Σg. We will first apply various cyclic group actions on Σg for g>0, and then extend it diagonally to the product manifolds ΣgxΣg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2.
Link Floer homology is a powerful invariant of links due to Ozsváth and Szabó. One of its most striking properties is that it detects each link's Thurston norm, a result also due to Ozsváth and Szabó. In this talk I will discuss generalizations of this result to the context of 4-ended tangles, as well as some tangle detection results. This is joint work in progress with Subhankar Dey and Claudius Zibrowius.
We give an overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic. In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g.
We give an overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic. In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g.
TBA
In the past few years there have been a host of remarkable topological results arising from considering "real" versions of various gauge and Floer-theoretic invariants of three- and four-dimensional manifolds equipped with involutions.
The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, and its behavior under gluing. I will also talk about a categorical framework for computing skein lasagna modules of closed 4-manifolds via trisection, as well as an extended 4d TQFT based on skein lasagna theory. This is joint work with Sarah Blackwell and Slava Krushkal.
